Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 36, 2011, 353-392
Mersin University, Faculty of Literature and Science
Department of Mathematics
33342 Mersin, Turkey; fabdul 'at' mersin.edu.tr
National Academy of Sciences of Ukraine,
Institute of Applied Mathematics and Mechanics
Roza Luxemburg Str. 74, 83 114, Donetsk, Ukraine;
aleksdov 'at' mail.ru
Mersin University, Faculty of Literature and Science
Department of Mathematics
33342 Mersin, Turkey; mkucukaslan@mersin.edu.tr
Abstract. We find necessary and sufficient conditions for an arbitrary metric space X to have a unique pretangent space at a marked point a \in X. Applying this general result we show that each logarithmic spiral has a unique pretangent space at the asymptotic point. Unbounded multiplicative subgroups of C* = C \ {0} having unique pretangent spaces at zero are characterized as lying either on the positive real semiaxis or on logarithmic spirals. Our general uniqueness conditions in the case X \subseteq R make it also possible to characterize the points of the ternary Cantor set having unique pretangent spaces.
2010 Mathematics Subject Classification: Primary 54E35.
Key words: Metric spaces, pretangent spaces, uniqueness of pretangent metric spaces, logarithmic spiral, Cantor set.
Reference to this article: F. Abdullayev, O. Dovgoshey and M. Kücükaslan: Metric spaces with unique pretangent spaces. Conditions of the uniqueness. Ann. Acad. Sci. Fenn. Math. 36 (2011), 353-392.
doi:10.5186/aasfm.2011.3623
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